How to Report the Friedman Test in APA 7

Friedman is the non-parametric counterpart of the repeated-measures ANOVA, and manuscripts that use it tend to fail in the same two places: the omnibus statistic arrives without an effect size, and a string of Wilcoxon tests follows with no correction at all, so the global test was conservative and everything after it is not. Both problems are cheap to fix, and this is the order the numbers go in. The wider map is in how to report non-parametric tests.

Which numbers go in

Four things: a description of each condition with median and interquartile range, the χ² statistic with its degrees of freedom and the N, the p value and Kendall's W as the effect size. On one line: χ²(3, N = 26) = 22.89, p < .001, W = .29.

Degrees of freedom are k − 1, where k is the number of repeated conditions. Three time points give df = 2; four conditions give df = 3. This catches out anyone arriving from ANOVA, where the error degrees of freedom depend on sample size. Here they do not: the N enters the statistic itself, which is exactly why it is declared inside the brackets, as it is for a contingency-table chi-square.

The χ is set upright, not italicized, because it is a Greek letter, the same way η² and β are. What is italicized is N, p and W. Kendall's W cannot exceed 1, so it takes no leading zero.

Describe the conditions with the median and the interquartile range. If you gave up the mean because the data are ordinal or badly skewed, reporting M and SD two lines later contradicts you in writing. Many authors also include the mean ranks the software prints, which are what the test actually compares: they live on a 1-to-k scale and always average to (k + 1)/2, so they are not scores and should not be read as such.

Kendall's W, the effect size reviewers ask for

Kendall's W is the chi-square divided by N(k − 1). For the example above: 22.89 / (26 × 3) = .29. You do not need extra software, only the statistic, the sample size and the number of conditions, which means there is no excuse for leaving it out.

It runs from 0 to 1 and reads as the extent to which participants rank the conditions in the same order. A 0 means everyone orders them differently; a 1 means everyone agrees on a single ordering. There is no canonical Cohen-style table for W: the usual reading borrows the benchmarks for r (around .10 small, .30 medium, .50 large), so present it as a working convention rather than an established cut-off.

It is the same coefficient of concordance used for agreement between judges, which is why the block design maps onto it so cleanly: if your participants are raters and your conditions are the objects they score, W tells you how much they converge. Once the line is written, the APA 7 formatter will check the italics, the decimals and the leading zeros for you.

Post hoc: pairwise Wilcoxon with a correction

A significant Friedman tells you that the conditions differ somewhere, not where. What follows are pairwise Wilcoxon signed-rank tests, and the number of comparisons is k(k − 1)/2: three comparisons with three conditions, six with four. Under Bonferroni the threshold moves from .05 to .017 in the first case and to .008 in the second. Bonferroni is the correction everyone recognizes and also the most conservative one: Holm gives the same guarantee with more power and is accepted without argument, so with six comparisons or more it is worth the extra sentence.

There are two correct ways to write this up, you only need one, and you must say which: compare the raw p against the adjusted alpha, or multiply each p by the number of comparisons and compare it against .05. The decision is identical. What does not work is leaving it implicit, because the reader cannot tell whether the .013 you flagged as significant survived your correction or only survived .05.

The warning that saves a round of review: if you used the automatic pairwise comparisons in the non-parametric menu of SPSS, the adjusted significance column already has Bonferroni applied. Correcting it again by hand is a double correction and it will bury real effects. Those automatic comparisons are also not Wilcoxon tests — they are Dunn-type contrasts on the mean ranks. Both routes are defensible; mixing them inside one paragraph is not.

Every comparison carries its own effect size, the familiar r = |z| / √N. Here N is the number of observations entering that comparison, twice the number of pairs: 26 participants measured twice give N = 52. Some authors use the number of pairs instead, which multiplies r by √2 — in the example below, an r of .55 would become .77 with the data untouched — so state which one you used.

A worked example you can copy

“Self-reported cognitive load differed across the four listening conditions: silence, Mdn = 3.0, IQR = 2.0–4.0; white noise, Mdn = 4.0, IQR = 3.0–5.0; instrumental music, Mdn = 5.0, IQR = 4.0–6.0; speech babble, Mdn = 6.0, IQR = 5.0–7.0. A Friedman test indicated a significant effect of condition, χ²(3, N = 26) = 22.89, p < .001, W = .29, with mean ranks of 1.65, 2.31, 2.73 and 3.31. Pairwise Wilcoxon signed-rank tests were run for the six contrasts and the p values were Bonferroni-adjusted. Cognitive load was higher under speech babble than under silence, z = −3.94, adjusted p < .001, r = .55, and higher under instrumental music than under silence, z = −3.34, adjusted p = .005, r = .46, and higher under speech babble than under white noise, z = −3.06, adjusted p = .013, r = .42. The remaining three contrasts were not significant (adjusted p > .17). Rank-based tests were used because the load ratings were ordinal and clearly skewed in every condition.”

Notice the order: description, omnibus test with its effect size, pairwise comparisons with the correction named out loud, and a closing sentence that justifies why this test rather than a repeated-measures ANOVA. That last sentence is the one most often missing and it costs nothing to write.

The paragraph also says something the numbers alone do not. Speech babble is the condition that stands apart, and the neighboring conditions do not separate from each other — that reading comes from which contrasts survived the correction, not from medians that happen to look different.

If what you have in front of you is an SPSS output with its ranks table and its test statistics table and you cannot tell which row is which, paste it into the output interpreter and come out with the sentence already assembled.

When Friedman is the wrong test

With only two conditions. Use the Wilcoxon signed-rank test directly. Friedman with k = 2 will still run, but it collapses into the sign test: with two conditions the only available ranks are 1 and 2, so the test counts who went up and who went down and throws away the size of each difference. Wilcoxon uses that size, which is why it has more power.

With missing data. Friedman works on complete cases: a participant who missed a single measurement is dropped from the analysis entirely. Across four conditions with any attrition, that can silently remove a third of your sample. When that is the real problem, mixed models are the sensible route, because they use whatever information each person contributed.

As a way of dodging sphericity. This is the genuinely good argument for Friedman, so use it properly: the test assumes neither normality nor sphericity, so there is no Mauchly test and no Greenhouse-Geisser correction to report. What it will not do is rescue a badly specified design, and a repeated-measures ANOVA with its correction applied remains more powerful whenever the data allow it.

By default, without looking at the data. Taking the non-parametric route every time just in case costs power, and in a small sample that loss is precisely what you cannot afford. The decision is made beforehand, by looking at the distribution, and it is explained in a single sentence in the text.

Frequently asked questions

How do you report a Friedman test in APA format?

With the chi-square statistic, its degrees of freedom and the N inside the brackets, the p value and an effect size: χ²(3, N = 26) = 22.89, p < .001, W = .29. Medians and interquartile ranges for each condition come first, and the corrected pairwise comparisons come after.

What are the degrees of freedom for the Friedman test?

They are k − 1, where k is the number of repeated conditions or time points, not the number of participants. Three time points give df = 2 and four conditions give df = 3. Sample size does not enter the degrees of freedom; it enters the statistic, which is why it is declared inside the brackets.

What effect size do you report with the Friedman test?

Kendall's W, obtained by dividing the chi-square by N(k − 1). It runs from 0 to 1, is italicized and takes no leading zero. There is no established benchmark table for W: the common reading borrows the values used for r, roughly .10 small, .30 medium and .50 large, and is best presented as a convention.

What post hoc test follows a significant Friedman test?

Pairwise Wilcoxon signed-rank tests with a multiple-comparison correction. Three conditions give three comparisons and an alpha of .017; four conditions give six comparisons and an alpha of .008. If you use the automatic pairwise comparisons in SPSS, the adjusted significance column already has Bonferroni applied, so do not correct it a second time.

Do you report means or medians with the Friedman test?

Medians with the interquartile range. If you chose a rank-based test it is because the mean does not describe your data well, so describing the conditions with means and standard deviations is incoherent and reviewers flag it. Many authors also include the mean ranks, which sit on a 1-to-k scale and are what the test actually compares.

Is the Friedman test the same as a repeated measures ANOVA?

No. Friedman ranks the scores within each participant, so it assumes neither normality nor sphericity and needs no Mauchly test or Greenhouse-Geisser correction. In exchange it loses power when the ANOVA assumptions hold, and it drops any participant with one missing measurement, which a mixed model does not.

If you are building the results section of a thesis or a paper and you are not certain the test you ran was the right one, that doubt is cheaper to settle before you write than after the manuscript comes back. Our statistical consultancy reviews the full analysis plan and puts in writing which test answers which question, and how each one should be reported.